Integral approximation method popular in condensed matter physics
A Sommerfeld expansion is an approximation method developed by Arnold Sommerfeld for a certain class of integrals which are common in condensed matter and statistical physics. Physically, the integrals represent statistical averages using the Fermi–Dirac distribution.
When the inverse temperature
is a large quantity, the integral can be expanded[1][2] in terms of
as
![{\displaystyle \int _{-\infty }^{\infty }{\frac {H(\varepsilon )}{e^{\beta (\varepsilon -\mu )}+1}}\,\mathrm {d} \varepsilon =\int _{-\infty }^{\mu }H(\varepsilon )\,\mathrm {d} \varepsilon +{\frac {\pi ^{2}}{6}}\left({\frac {1}{\beta }}\right)^{2}H^{\prime }(\mu )+O\left({\frac {1}{\beta \mu }}\right)^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f983830f3d4215cb10eb0d9cd34ba5df0b96e227)
where
is used to denote the derivative of
evaluated at
and where the
notation refers to limiting behavior of order
. The expansion is only valid if
vanishes as
and goes no faster than polynomially in
as
. If the integral is from zero to infinity, then the integral in the first term of the expansion is from zero to
and the second term is unchanged.
Application to the free electron model
Integrals of this type appear frequently when calculating electronic properties, like the heat capacity, in the free electron model of solids. In these calculations the above integral expresses the expected value of the quantity
. For these integrals we can then identify
as the inverse temperature and
as the chemical potential. Therefore, the Sommerfeld expansion is valid for large
(low temperature) systems.
Derivation to second order in temperature
We seek an expansion that is second order in temperature, i.e., to
, where
is the product of temperature and the Boltzmann constant. Begin with a change variables to
:
![{\displaystyle I=\int _{-\infty }^{\infty }{\frac {H(\varepsilon )}{e^{\beta (\varepsilon -\mu )}+1}}\,\mathrm {d} \varepsilon =\tau \int _{-\infty }^{\infty }{\frac {H(\mu +\tau x)}{e^{x}+1}}\,\mathrm {d} x\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac7fd72e0bf8e8e9c946d0859d51ad59e6bf5b8d)
Divide the range of integration,
, and rewrite
using the change of variables
:
![{\displaystyle I=\underbrace {\tau \int _{-\infty }^{0}{\frac {H(\mu +\tau x)}{e^{x}+1}}\,\mathrm {d} x} _{I_{1}}+\underbrace {\tau \int _{0}^{\infty }{\frac {H(\mu +\tau x)}{e^{x}+1}}\,\mathrm {d} x} _{I_{2}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/300eb0420614ce00ae78eb5800e31b989a8b5b41)
![{\displaystyle I_{1}=\tau \int _{-\infty }^{0}{\frac {H(\mu +\tau x)}{e^{x}+1}}\,\mathrm {d} x=\tau \int _{0}^{\infty }{\frac {H(\mu -\tau x)}{e^{-x}+1}}\,\mathrm {d} x\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c446d6e21bb52ecb8dfb2e6e430fd9fa9231b22d)
Next, employ an algebraic 'trick' on the denominator of
,
![{\displaystyle {\frac {1}{e^{-x}+1}}=1-{\frac {1}{e^{x}+1}}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f532aa437ec686b81dae0b0727c69bb24f05d489)
to obtain:
![{\displaystyle I_{1}=\tau \int _{0}^{\infty }H(\mu -\tau x)\,\mathrm {d} x-\tau \int _{0}^{\infty }{\frac {H(\mu -\tau x)}{e^{x}+1}}\,\mathrm {d} x\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3eb7e3223d027e0929c1de6dd095838638a42591)
Return to the original variables with
in the first term of
. Combine
to obtain:
![{\displaystyle I=\int _{-\infty }^{\mu }H(\varepsilon )\,\mathrm {d} \varepsilon +\tau \int _{0}^{\infty }{\frac {H(\mu +\tau x)-H(\mu -\tau x)}{e^{x}+1}}\,\mathrm {d} x\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8d10947fbca99e9c5f9c8ba6e12f1ef598d952)
The numerator in the second term can be expressed as an approximation to the first derivative, provided
is sufficiently small and
is sufficiently smooth:
![{\displaystyle \Delta H=H(\mu +\tau x)-H(\mu -\tau x)\approx 2\tau xH'(\mu )+\cdots \,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9fae98dfb0cded51e475ff09ff57d4b9dafdd59)
to obtain,
![{\displaystyle I=\int _{-\infty }^{\mu }H(\varepsilon )\,\mathrm {d} \varepsilon +2\tau ^{2}H'(\mu )\int _{0}^{\infty }{\frac {x\mathrm {d} x}{e^{x}+1}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92607fc98eed201cf9ce556b45d6d91b8f29d908)
The definite integral is known[3] to be:
.
Hence,
![{\displaystyle I=\int _{-\infty }^{\infty }{\frac {H(\varepsilon )}{e^{\beta (\varepsilon -\mu )}+1}}\,\mathrm {d} \varepsilon \approx \int _{-\infty }^{\mu }H(\varepsilon )\,\mathrm {d} \varepsilon +{\frac {\pi ^{2}}{6\beta ^{2}}}H'(\mu )\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8bb47dc5c8caa4b62a99e1cd2c3bb5610d8793b)
Higher order terms and a generating function
We can obtain higher order terms in the Sommerfeld expansion by use of a generating function for moments of the Fermi distribution. This is given by
![{\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi }}e^{\tau \epsilon /2\pi }\left\{{\frac {1}{1+e^{\beta (\epsilon -\mu )}}}-\theta (-\epsilon )\right\}={\frac {1}{\tau }}\left\{{\frac {({\frac {\tau T}{2}})}{\sin({\frac {\tau T}{2}})}}e^{\tau \mu /2\pi }-1\right\},\quad 0<\tau T/2\pi <1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5284e2e04749c0926b29c48be39ab41d589d8964)
Here
and Heaviside step function
subtracts the divergent zero-temperature contribution. Expanding in powers of
gives, for example [4]
![{\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi }}\left\{{\frac {1}{1+e^{\beta (\epsilon -\mu )}}}-\theta (-\epsilon )\right\}=\left({\frac {\mu }{2\pi }}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5f87d5cb601e027145fd5d96538a9f63885a2c7)
![{\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi }}\left({\frac {\epsilon }{2\pi }}\right)\left\{{\frac {1}{1+e^{\beta (\epsilon -\mu )}}}-\theta (-\epsilon )\right\}={\frac {1}{2!}}\left({\frac {\mu }{2\pi }}\right)^{2}+{\frac {T^{2}}{4!}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63889301fa874652fb1e78c1bb3da85c5b63ed97)
![{\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi }}{\frac {1}{2!}}\left({\frac {\epsilon }{2\pi }}\right)^{2}\left\{{\frac {1}{1+e^{\beta (\epsilon -\mu )}}}-\theta (-\epsilon )\right\}={\frac {1}{3!}}\left({\frac {\mu }{2\pi }}\right)^{3}+\left({\frac {\mu }{2\pi }}\right){\frac {T^{2}}{4!}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab0c3b33b04517dff8a7f98705377e0508771a33)
![{\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi }}{\frac {1}{3!}}\left({\frac {\epsilon }{2\pi }}\right)^{3}\left\{{\frac {1}{1+e^{\beta (\epsilon -\mu )}}}-\theta (-\epsilon )\right\}={\frac {1}{4!}}\left({\frac {\mu }{2\pi }}\right)^{4}+{\frac {1}{2!}}\left({\frac {\mu }{2\pi }}\right)^{2}{\frac {T^{2}}{4!}}+{\frac {7}{8}}{\frac {T^{4}}{6!}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ffd6b9d1a636fde5c99e13ccc7fa43f5e7495e9)
![{\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi }}{\frac {1}{4!}}\left({\frac {\epsilon }{2\pi }}\right)^{4}\left\{{\frac {1}{1+e^{\beta (\epsilon -\mu )}}}-\theta (-\epsilon )\right\}={\frac {1}{5!}}\left({\frac {\mu }{2\pi }}\right)^{5}+{\frac {1}{3!}}\left({\frac {\mu }{2\pi }}\right)^{3}{\frac {T^{2}}{4!}}+\left({\frac {\mu }{2\pi }}\right){\frac {7}{8}}{\frac {T^{4}}{6!}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c526b7a329ee5fbe3e7fac2f4f588a303bccf5d8)
![{\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi }}{\frac {1}{5!}}\left({\frac {\epsilon }{2\pi }}\right)^{5}\left\{{\frac {1}{1+e^{\beta (\epsilon -\mu )}}}-\theta (-\epsilon )\right\}={\frac {1}{6!}}\left({\frac {\mu }{2\pi }}\right)^{6}+{\frac {1}{4!}}\left({\frac {\mu }{2\pi }}\right)^{4}{\frac {T^{2}}{4!}}+{\frac {1}{2!}}\left({\frac {\mu }{2\pi }}\right)^{2}{\frac {7}{8}}{\frac {T^{4}}{6!}}+{\frac {31}{24}}{\frac {T^{6}}{8!}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b3dd4499f1d5bbf459edfaa3821fa28d59e4629)
A similar generating function for the odd moments of the Bose function is
![{\displaystyle \int _{0}^{\infty }{\frac {d\epsilon }{2\pi }}\sinh(\epsilon \tau /\pi ){\frac {1}{e^{\beta \epsilon }-1}}={\frac {1}{4\tau }}\left\{1-{\frac {\tau T}{\tan \tau T}}\right\},\quad 0<\tau T<\pi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d1964989044d888bd479ac4d6e0b1bc7551b957)
Notes
- ^ Ashcroft & Mermin 1976, p. 760.
- ^ Fabian, J. "Sommerfeld's expansion" (PDF). Universitaet Regensburg. Retrieved 2016-02-08.
- ^ "Definite integrals containing exponential functions". SOS Math. Retrieved 2016-02-08.
- ^ R. Loganayagam, P. Surówka (2012). "Anomaly/Transport in an Ideal Weyl gas". JHEP. 2012 (4): 2012:97. arXiv:1201.2812. Bibcode:2012JHEP...04..097L. CiteSeerX 10.1.1.761.5605. doi:10.1007/JHEP04(2012)097. S2CID 118841274.
References
- Sommerfeld, A. (1928). "Zur Elektronentheorie der Metalle auf Grund der Fermischen Statistik". Zeitschrift für Physik. 47 (1–2): 1–3. Bibcode:1928ZPhy...47....1S. doi:10.1007/BF01391052. S2CID 116911592.
- Ashcroft, Neil W.; Mermin, N. David (1976). Solid State Physics. Thomson Learning. p. 760. ISBN 978-0-03-083993-1.